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15k is a healthy budget, though I'd rather find a more sorted (manual) M5 with most work done to it already for 25k than a 10k 8 owner craigslist special dealing with who knows how many headaches down the road.
For a beater M5 you're prob going to have to catch up on a lot of deferred maintenance. All your fluids, spark plugs, and almost guarantee rotors/pads are easy though you'll need a computer to bleed the brakes. Maybe ignition coils, looking about time for 2 throttle body actuators and 2 idle air actuators. Need to do bearings, which while you're there might as well do the vanos high pressure line and motor mounts. I'd also look at a set of decent tires and any worn suspension bits. If you need to drop the SMG for work, look into doing the rear main seal on the diff as they seem to go with age.
I'm sure some else can chime in with what the SMG might need, I'm a snobby manual elitist with no 1st hand ownership experience.
With buying your own parts being smart where you get them, and having all that done at a good german indy; I feel pretty confident all that work could get done for under 15k at least where I am.
91 should be the lowest that you put in car. If that's all you get in your area I get it, but go 93 if you can.
Power also doesn't come cheap, headers, exhaust, under drive pulley, tune, and intake scoops may net you 70+ horses on a tired motor but at over 6k installed. Any more power and you're looking at a stroker build or supercharging.
I'd make sure it runs right before going down that rabbit hole.
07 6 speed E60 M5
12 6 speed E93 M3